Poisson Lie algebroid in nLab

Contents

Contents

Idea

A Poisson Lie algebroid on a manifold XX is a Lie algebroid on XX naturally defined from and defining the structure of a Poisson manifold on XX.

This is the degree-1 example of a tower of related concepts, described at n-symplectic manifold.

Definition

Let π∈Γ(⋀ 2TX)\pi \in \Gamma(\Wedge^2 TX) be a Poisson manifold structure, incarnated as a Poisson tensor.

As vector-bundle with anchor

In terms of the vector-bundle-with anchor definition of Lie algebroid the Poisson Lie algebroid 𝔓(X,π)\mathfrak{P}(X,\pi) corresponding to π\pi is the cotangent bundle

T *X →π(−) TX ↘ ↙ X \array{ T^* X &&\stackrel{\pi(-)}{\to}&& T X \\ & \searrow && \swarrow \\ && X }

equipped with the anchor map that sends a differential 1-form α\alpha to the vector obtained by contraction with the Poisson bivector π:α↦π(α,−)\pi \colon \alpha \mapsto \pi(\alpha,-).

The Lie bracket [−,−]:Γ(T *X)∧Γ(T *X)→Γ(T *X)[-,-] : \Gamma(T^* X) \wedge \Gamma(T^* X) \to \Gamma(T^* X) is given by

[α,β]≔ℒ π(α)β−ℒ π(β)α−d dR(π(α,β)), [\alpha,\beta] \coloneqq \mathcal{L}_{\pi(\alpha)} \beta - \mathcal{L}_{\pi(\beta)} \alpha - d_{dR}(\pi(\alpha,\beta))\,,

where ℒ\mathcal{L} denotes the Lie derivative and d dRd_{dR} the de Rham differential. This is the unique Lie algebroid bracket on T *X→πTXT^* X \stackrel{\pi}{\to} T X which is given on exact differential 1-forms by

[d dRf,d dRg]=d dR{f,g} [d_{dR} f, d_{dR} g] = d_{dR} \{f,g\}

for all f,g∈C ∞(X)f,g \in C^\infty(X). On a coordinate patch this reduces to

[dx i,dx j]=d dRπ ij [d x^i , d x^j] = d_{dR} \pi^{i j}

for {x i}\{x^i\} the coordinate functios and {π ij}\{\pi^{i j}\} the components of the Poisson tensor in these coordinates.

Chevalley-Eilenberg algebra

We describe the Chevalley-Eilenberg algebra of the Poisson Lie algebra given by π\pi, which defines it dually.

Notice that π\pi is an element of degree 2 in the exterior algebra ∧ •Γ(TX)\wedge^\bullet \Gamma(T X) of multivector fields on XX. The Lie bracket on tangent vectors in Γ(TX)\Gamma(T X) extends to a bracket [−,−] Sch[-,-]_{Sch} on multivector field, the Schouten bracket. The defining property of the Poisson structure π\pi is that

[π,π] Sch=0. [\pi,\pi]_{Sch} = 0 \,.

This makes

d CE(𝔓(X,π)):=[π,−]:CE(𝔓(X,π))→CE(𝔓(X,π))) d_{CE(\mathfrak{P}(X,\pi))} := [\pi, -] : CE(\mathfrak{P}(X,\pi)) \to CE(\mathfrak{P}(X,\pi)))

into a differential of degree +1 on multivector fields, that squares to 0. We write CE(𝔓(X,π))CE(\mathfrak{P}(X,\pi)) for the exterior algebra equipped with this differential.

More explicitly, let {x i}:X→ℝ dimX\{x^i\} : X \to \mathbb{R}^{dim X} be a coordinate patch. Then the differential of CE(𝔓(X,π))CE(\mathfrak{P}(X,\pi)) is given by

d 𝔓(X,π):x i↦2π ij∂ j d_{\mathfrak{P}(X,\pi)} : x^i \mapsto 2 \pi^{i j} \partial_j

d 𝔓(X,π):∂ i↦.... d_{\mathfrak{P}(X,\pi)} : \partial_i \mapsto ... \,.

Properties

Cohomology and Chern-Simons elements

We discuss aspects of the ∞-Lie algebroid cohomology of Poisson Lie algebroids 𝔓(X,π)\mathfrak{P}(X,\pi). This is equivalently called Poisson cohomology (see there for details).

We shall be lazy (and follow tradition) and write the following formulas in a local coordinate patch {x i}\{x^i\} for XX.

Then the Chevalley-Eilenberg algebra CE(𝔓(X,π))CE(\mathfrak{P}(X,\pi)) is generated from the x ix^i and the ∂ i\partial_i, and the Weil algebra W(𝔓(X,π))W(\mathfrak{P}(X,\pi)) is generated from x ix^i, ∂ i\partial_i and their shifted partners, which we shall write dx i\mathbf{d} x^i and d∂ i\mathbf{d}\partial_i. The differential on the Weil algebra we may then write

d W(𝔓(X,π))=[π,−] Sch+d. d_{W(\mathfrak{P}(X,\pi))} = [\pi,-]_{Sch} + \mathbf{d} \,.

Notice that π∈CE(𝔓(X,π))\pi \in CE(\mathfrak{P}(X,\pi)) is a Lie algebroid cocycle, since

d CE(𝔓(X,π))π=[π,π] Sch=0. d_{CE(\mathfrak{P}(X,\pi))} \pi = [\pi,\pi]_{Sch} = 0 \,.

Lagrangian submanifolds and coisotropic submanifolds

The Lagrangian dg-submanifolds (see there for more) of a Poisson Lie algebroid correspond to the coisotropic submanifolds of the corresponding Poisson manifold.

• Under Lie integration a Poisson Lie algebroid is supposed to yield a symplectic groupoid.

• There is a formulation of Legendre transformation in terms of Lie algebroid.

• symplectic Lie n-algebroid

  • symplectic manifold

  • Poisson Lie algebroid

  • Courant algebroid

• Hopf algebroid (appears as a deformation quantization of a Poisson-Lie algebroid)

Examples

• For the simple but important special case of Lie-Poisson structure see there at Poisson-Lie algebroid cohomology.

References

One of the earliest reference seems to be

• Ted Courant, Tangent Lie algebroid (pdf)

A review is for instance in

• Dmitry Roytenberg, appendix A Courant algebroids, derived brackets and even symplectic supermanifolds (arXiv:math/9910078)

The H-cohomology of the graded symplectic form of a Poisson Lie algebroid, regarded a a symplectic Lie n-algebroid, is considered in

• Pavol Ševera, p. 1 of On the origin of the BV operator on odd symplectic supermanifolds, Lett Math Phys (2006) 78: 55. (arXiv:0506331)